How to find the length of an arc - Intermediate Geometry

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Question

In the circle above, the angle A in radians is $\frac{\pi}{4}$

What is the length of arc A?

Answer

Circumference of a Circle = $2\pi r$

Arc Length

$=\frac{Angle A}{2\pi}\ast2\pi r$

$=\frac{\frac{\pi}{4}}{2\pi}\ast2\pi r$

$=\frac{\pi}{(4)2\pi}\ast2\pi r$

$=\frac{1}{8}\ast2\pi r=\frac{\pi r}{4}$

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Question

If the area of a circle is 1.44 $\pi$ , what is its circumference?

Answer

The answer is $2.4 \pi$ .

Utilizing the formula for area of a circle $\left( \pi r^{2}\right)$ , you would plug in the answer for area as

$1.44\pi$ = $\pi r^{2}$

Divide both sides by $\pi$ .

Then square root both sides to get

Then plug in 1.2 for in the equation for circumference for a circle, $2\pi r$ . Thus

$C = 2.4\pi$

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Question

Find the perimeter around the following semicircle.

Answer

The answer is $9\pi + 18cm$ .

First, you would need to find the radius of the semi-circle. 18 divided by 2 results in 9 cm for the radius. Then you would take the formula for finding circumference $C=2\pi r$ and plug in to get

$C=18\pi$ .

Then you would divide that result by 2 to get $9\pi$ since it is a semicircle. Lastly you would add 18 cm to $9\pi$ because the perimeter is the sum of the semicircle and the diameter. Remember that they are not like terms.

If you chose $9\pi$ , you forgot to include the diameter.

If you chose $27\pi$ , you added and , but they are not like terms.

If you chose $18\pi + 18$ , remember that you only need half of the circumference.

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Question

A sector of a circle with radius of feet has an area of $\small 36\pi$ square feet. Find the length of the arc of the sector.

Answer

We begin with the formula for the area of a sector

$\small S=\frac{m}{360}\pi r^2$

where $\small m$ is the measure of the central angle in degrees and $\small r$ is the radius.

Substituting what we know gives

$\small 36\pi=\frac{m}{360}\pi (12^2)$

$\small 36\pi=\frac{4\pi m}{10}$

$\small 360\pi=4\pi m$

$\small 90=m$

Therefore our central angle is $\small 90^\circ$

We then turn to our formula for arc length.

$\small s=\frac{m}{360}2\pi r$

Substituting gives

$\small s=\frac{90}{360}(2\pi)(12)=\frac{1}{4}(24\pi)=6\pi$

Therefore, our arc length is $\small 6\pi,ft.$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{120}{360}\times 2\pi\times2=\frac{4}{3}\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measurement of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{240}{360}\times 2\pi\times 3=4\pi$

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Question

The radius of a circle is . Find the length of an arc that has a measurement of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{45}{360}\times 2\pi\times 4=\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{150}{360}\times 2\pi\times 5=\frac{25}{6}\pi$

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Question

The radius of a circle is . Find the length of an arc that has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{95}{360}\times 2\pi\times 6=\frac{19}{6}\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{30}{360}\times 2\pi\times 6=\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{60}{360}\times 2\pi\times 8=\frac{2}{3}\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{200}{360}\times 2\pi\times 9=10\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{40}{360}\times 2\pi\times 10=\frac{20}{9}\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{130}{360}\times 2\pi\times 12=\frac{26}{3}\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{180}{360}\times 2\pi\times 12=12\pi$

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Question

The radius of a circle is . Find the length of an arc if it has a measure of degrees.

Answer

Recall that the length of an arc is merely a part of the circle's circumference.

We can then write the following equation to find the length of an arc:

$\text{Arc Length}=\frac{\text{Arc angle measure}}{360}\times2\pi\times \text{radius}$

Plug in the values of the arc angle measure and the radius to find the length of the arc.

$\text{Arc Length}=\frac{210}{360}\times 2\pi\times 1=\frac{7}{6}\pi$

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Question

Find the length of the arc if the radius of a circle is and the measure of the central angle is degrees.

Answer

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

$\text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{55}{360}\times 2\pi \times 12$

Solve.

$\text{Length of Arc}=\frac{11}{3}\pi$

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Question

Find the length of an arc if the radius of the circle is and the measurement of the central angle is degrees.

Answer

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

$\text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{65}{360}\times 2\pi \times 13$

Solve.

$\text{Length of Arc}=\frac{169}{36}\pi$

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Question

Find the length of an arc if the radius of the circle is and the measurement of the central angle is degrees.

Answer

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

$\text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{75}{360}\times 2\pi \times 14$

Solve.

$\text{Length of Arc}=\frac{35}{6}\pi$

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Question

Find the length of an arc if the radius of the circle is and the measurement of the central angle is degrees.

Answer

An arc is just a piece—or a fraction—of a circle's circumference. Use the following formula to find the length of an arc:

$\text{Length of Arc}=\frac{\text{Measure of central angle}}{360}\times2\pi\times \text{radius}$

Substitute in the given values for the central angle and the radius.

$\text{Length of Arc}=\frac{80}{360}\times 2\pi \times 15$

Solve.

$\text{Length of Arc}=\frac{20}{3}\pi$

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